Integrand size = 24, antiderivative size = 269 \[ \int \frac {\sqrt {a+b x^3} \left (A+B x^3\right )}{x^{13/2}} \, dx=\frac {2 (2 A b-11 a B) \sqrt {a+b x^3}}{55 a x^{5/2}}-\frac {2 A \left (a+b x^3\right )^{3/2}}{11 a x^{11/2}}-\frac {3^{3/4} b (2 A b-11 a B) \sqrt {x} \left (\sqrt [3]{a}+\sqrt [3]{b} x\right ) \sqrt {\frac {a^{2/3}-\sqrt [3]{a} \sqrt [3]{b} x+b^{2/3} x^2}{\left (\sqrt [3]{a}+\left (1+\sqrt {3}\right ) \sqrt [3]{b} x\right )^2}} \operatorname {EllipticF}\left (\arccos \left (\frac {\sqrt [3]{a}+\left (1-\sqrt {3}\right ) \sqrt [3]{b} x}{\sqrt [3]{a}+\left (1+\sqrt {3}\right ) \sqrt [3]{b} x}\right ),\frac {1}{4} \left (2+\sqrt {3}\right )\right )}{55 a^{4/3} \sqrt {\frac {\sqrt [3]{b} x \left (\sqrt [3]{a}+\sqrt [3]{b} x\right )}{\left (\sqrt [3]{a}+\left (1+\sqrt {3}\right ) \sqrt [3]{b} x\right )^2}} \sqrt {a+b x^3}} \] Output:
2/55*(2*A*b-11*B*a)*(b*x^3+a)^(1/2)/a/x^(5/2)-2/11*A*(b*x^3+a)^(3/2)/a/x^( 11/2)-1/55*3^(3/4)*b*(2*A*b-11*B*a)*x^(1/2)*(a^(1/3)+b^(1/3)*x)*((a^(2/3)- a^(1/3)*b^(1/3)*x+b^(2/3)*x^2)/(a^(1/3)+(1+3^(1/2))*b^(1/3)*x)^2)^(1/2)*In verseJacobiAM(arccos((a^(1/3)+(1-3^(1/2))*b^(1/3)*x)/(a^(1/3)+(1+3^(1/2))* b^(1/3)*x)),1/4*6^(1/2)+1/4*2^(1/2))/a^(4/3)/(b^(1/3)*x*(a^(1/3)+b^(1/3)*x )/(a^(1/3)+(1+3^(1/2))*b^(1/3)*x)^2)^(1/2)/(b*x^3+a)^(1/2)
Result contains higher order function than in optimal. Order 5 vs. order 4 in optimal.
Time = 10.11 (sec) , antiderivative size = 80, normalized size of antiderivative = 0.30 \[ \int \frac {\sqrt {a+b x^3} \left (A+B x^3\right )}{x^{13/2}} \, dx=\frac {2 \sqrt {a+b x^3} \left (-5 A \left (a+b x^3\right )+\frac {(2 A b-11 a B) x^3 \operatorname {Hypergeometric2F1}\left (-\frac {5}{6},-\frac {1}{2},\frac {1}{6},-\frac {b x^3}{a}\right )}{\sqrt {1+\frac {b x^3}{a}}}\right )}{55 a x^{11/2}} \] Input:
Integrate[(Sqrt[a + b*x^3]*(A + B*x^3))/x^(13/2),x]
Output:
(2*Sqrt[a + b*x^3]*(-5*A*(a + b*x^3) + ((2*A*b - 11*a*B)*x^3*Hypergeometri c2F1[-5/6, -1/2, 1/6, -((b*x^3)/a)])/Sqrt[1 + (b*x^3)/a]))/(55*a*x^(11/2))
Time = 0.55 (sec) , antiderivative size = 265, normalized size of antiderivative = 0.99, number of steps used = 5, number of rules used = 4, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.167, Rules used = {955, 809, 851, 766}
Below are the steps used by Rubi to obtain the solution. The rule number used for the transformation is given above next to the arrow. The rules definitions used are listed below.
| \(\displaystyle \int \frac {\sqrt {a+b x^3} \left (A+B x^3\right )}{x^{13/2}} \, dx\) |
| \(\Big \downarrow \) 955 |
| \(\displaystyle -\frac {(2 A b-11 a B) \int \frac {\sqrt {b x^3+a}}{x^{7/2}}dx}{11 a}-\frac {2 A \left (a+b x^3\right )^{3/2}}{11 a x^{11/2}}\) |
| \(\Big \downarrow \) 809 |
| \(\displaystyle -\frac {(2 A b-11 a B) \left (\frac {3}{5} b \int \frac {1}{\sqrt {x} \sqrt {b x^3+a}}dx-\frac {2 \sqrt {a+b x^3}}{5 x^{5/2}}\right )}{11 a}-\frac {2 A \left (a+b x^3\right )^{3/2}}{11 a x^{11/2}}\) |
| \(\Big \downarrow \) 851 |
| \(\displaystyle -\frac {(2 A b-11 a B) \left (\frac {6}{5} b \int \frac {1}{\sqrt {b x^3+a}}d\sqrt {x}-\frac {2 \sqrt {a+b x^3}}{5 x^{5/2}}\right )}{11 a}-\frac {2 A \left (a+b x^3\right )^{3/2}}{11 a x^{11/2}}\) |
| \(\Big \downarrow \) 766 |
| \(\displaystyle -\frac {(2 A b-11 a B) \left (\frac {3^{3/4} b \sqrt {x} \left (\sqrt [3]{a}+\sqrt [3]{b} x\right ) \sqrt {\frac {a^{2/3}-\sqrt [3]{a} \sqrt [3]{b} x+b^{2/3} x^2}{\left (\sqrt [3]{a}+\left (1+\sqrt {3}\right ) \sqrt [3]{b} x\right )^2}} \operatorname {EllipticF}\left (\arccos \left (\frac {\left (1-\sqrt {3}\right ) \sqrt [3]{b} x+\sqrt [3]{a}}{\left (1+\sqrt {3}\right ) \sqrt [3]{b} x+\sqrt [3]{a}}\right ),\frac {1}{4} \left (2+\sqrt {3}\right )\right )}{5 \sqrt [3]{a} \sqrt {\frac {\sqrt [3]{b} x \left (\sqrt [3]{a}+\sqrt [3]{b} x\right )}{\left (\sqrt [3]{a}+\left (1+\sqrt {3}\right ) \sqrt [3]{b} x\right )^2}} \sqrt {a+b x^3}}-\frac {2 \sqrt {a+b x^3}}{5 x^{5/2}}\right )}{11 a}-\frac {2 A \left (a+b x^3\right )^{3/2}}{11 a x^{11/2}}\) |
Input:
Int[(Sqrt[a + b*x^3]*(A + B*x^3))/x^(13/2),x]
Output:
(-2*A*(a + b*x^3)^(3/2))/(11*a*x^(11/2)) - ((2*A*b - 11*a*B)*((-2*Sqrt[a + b*x^3])/(5*x^(5/2)) + (3^(3/4)*b*Sqrt[x]*(a^(1/3) + b^(1/3)*x)*Sqrt[(a^(2 /3) - a^(1/3)*b^(1/3)*x + b^(2/3)*x^2)/(a^(1/3) + (1 + Sqrt[3])*b^(1/3)*x) ^2]*EllipticF[ArcCos[(a^(1/3) + (1 - Sqrt[3])*b^(1/3)*x)/(a^(1/3) + (1 + S qrt[3])*b^(1/3)*x)], (2 + Sqrt[3])/4])/(5*a^(1/3)*Sqrt[(b^(1/3)*x*(a^(1/3) + b^(1/3)*x))/(a^(1/3) + (1 + Sqrt[3])*b^(1/3)*x)^2]*Sqrt[a + b*x^3])))/( 11*a)
Int[1/Sqrt[(a_) + (b_.)*(x_)^6], x_Symbol] :> With[{r = Numer[Rt[b/a, 3]], s = Denom[Rt[b/a, 3]]}, Simp[x*(s + r*x^2)*(Sqrt[(s^2 - r*s*x^2 + r^2*x^4)/ (s + (1 + Sqrt[3])*r*x^2)^2]/(2*3^(1/4)*s*Sqrt[a + b*x^6]*Sqrt[r*x^2*((s + r*x^2)/(s + (1 + Sqrt[3])*r*x^2)^2)]))*EllipticF[ArcCos[(s + (1 - Sqrt[3])* r*x^2)/(s + (1 + Sqrt[3])*r*x^2)], (2 + Sqrt[3])/4], x]] /; FreeQ[{a, b}, x ]
Int[((c_.)*(x_))^(m_.)*((a_) + (b_.)*(x_)^(n_))^(p_), x_Symbol] :> Simp[(c* x)^(m + 1)*((a + b*x^n)^p/(c*(m + 1))), x] - Simp[b*n*(p/(c^n*(m + 1))) I nt[(c*x)^(m + n)*(a + b*x^n)^(p - 1), x], x] /; FreeQ[{a, b, c}, x] && IGtQ [n, 0] && GtQ[p, 0] && LtQ[m, -1] && !ILtQ[(m + n*p + n + 1)/n, 0] && IntB inomialQ[a, b, c, n, m, p, x]
Int[((c_.)*(x_))^(m_)*((a_) + (b_.)*(x_)^(n_))^(p_), x_Symbol] :> With[{k = Denominator[m]}, Simp[k/c Subst[Int[x^(k*(m + 1) - 1)*(a + b*(x^(k*n)/c^ n))^p, x], x, (c*x)^(1/k)], x]] /; FreeQ[{a, b, c, p}, x] && IGtQ[n, 0] && FractionQ[m] && IntBinomialQ[a, b, c, n, m, p, x]
Int[((e_.)*(x_))^(m_.)*((a_) + (b_.)*(x_)^(n_))^(p_.)*((c_) + (d_.)*(x_)^(n _)), x_Symbol] :> Simp[c*(e*x)^(m + 1)*((a + b*x^n)^(p + 1)/(a*e*(m + 1))), x] + Simp[(a*d*(m + 1) - b*c*(m + n*(p + 1) + 1))/(a*e^n*(m + 1)) Int[(e *x)^(m + n)*(a + b*x^n)^p, x], x] /; FreeQ[{a, b, c, d, e, p}, x] && NeQ[b* c - a*d, 0] && (IntegerQ[n] || GtQ[e, 0]) && ((GtQ[n, 0] && LtQ[m, -1]) || (LtQ[n, 0] && GtQ[m + n, -1])) && !ILtQ[p, -1]
Result contains complex when optimal does not.
Time = 4.34 (sec) , antiderivative size = 745, normalized size of antiderivative = 2.77
| method | result | size |
| risch | \(-\frac {2 \sqrt {b \,x^{3}+a}\, \left (3 A b \,x^{3}+11 B a \,x^{3}+5 A a \right )}{55 x^{\frac {11}{2}} a}-\frac {6 b^{2} \left (2 A b -11 B a \right ) \left (\frac {\left (-a \,b^{2}\right )^{\frac {1}{3}}}{2 b}-\frac {i \sqrt {3}\, \left (-a \,b^{2}\right )^{\frac {1}{3}}}{2 b}\right ) \sqrt {\frac {\left (-\frac {3 \left (-a \,b^{2}\right )^{\frac {1}{3}}}{2 b}+\frac {i \sqrt {3}\, \left (-a \,b^{2}\right )^{\frac {1}{3}}}{2 b}\right ) x}{\left (-\frac {\left (-a \,b^{2}\right )^{\frac {1}{3}}}{2 b}+\frac {i \sqrt {3}\, \left (-a \,b^{2}\right )^{\frac {1}{3}}}{2 b}\right ) \left (x -\frac {\left (-a \,b^{2}\right )^{\frac {1}{3}}}{b}\right )}}\, {\left (x -\frac {\left (-a \,b^{2}\right )^{\frac {1}{3}}}{b}\right )}^{2} \sqrt {\frac {\left (-a \,b^{2}\right )^{\frac {1}{3}} \left (x +\frac {\left (-a \,b^{2}\right )^{\frac {1}{3}}}{2 b}+\frac {i \sqrt {3}\, \left (-a \,b^{2}\right )^{\frac {1}{3}}}{2 b}\right )}{b \left (-\frac {\left (-a \,b^{2}\right )^{\frac {1}{3}}}{2 b}-\frac {i \sqrt {3}\, \left (-a \,b^{2}\right )^{\frac {1}{3}}}{2 b}\right ) \left (x -\frac {\left (-a \,b^{2}\right )^{\frac {1}{3}}}{b}\right )}}\, \sqrt {\frac {\left (-a \,b^{2}\right )^{\frac {1}{3}} \left (x +\frac {\left (-a \,b^{2}\right )^{\frac {1}{3}}}{2 b}-\frac {i \sqrt {3}\, \left (-a \,b^{2}\right )^{\frac {1}{3}}}{2 b}\right )}{b \left (-\frac {\left (-a \,b^{2}\right )^{\frac {1}{3}}}{2 b}+\frac {i \sqrt {3}\, \left (-a \,b^{2}\right )^{\frac {1}{3}}}{2 b}\right ) \left (x -\frac {\left (-a \,b^{2}\right )^{\frac {1}{3}}}{b}\right )}}\, \operatorname {EllipticF}\left (\sqrt {\frac {\left (-\frac {3 \left (-a \,b^{2}\right )^{\frac {1}{3}}}{2 b}+\frac {i \sqrt {3}\, \left (-a \,b^{2}\right )^{\frac {1}{3}}}{2 b}\right ) x}{\left (-\frac {\left (-a \,b^{2}\right )^{\frac {1}{3}}}{2 b}+\frac {i \sqrt {3}\, \left (-a \,b^{2}\right )^{\frac {1}{3}}}{2 b}\right ) \left (x -\frac {\left (-a \,b^{2}\right )^{\frac {1}{3}}}{b}\right )}}, \sqrt {\frac {\left (\frac {3 \left (-a \,b^{2}\right )^{\frac {1}{3}}}{2 b}+\frac {i \sqrt {3}\, \left (-a \,b^{2}\right )^{\frac {1}{3}}}{2 b}\right ) \left (\frac {\left (-a \,b^{2}\right )^{\frac {1}{3}}}{2 b}-\frac {i \sqrt {3}\, \left (-a \,b^{2}\right )^{\frac {1}{3}}}{2 b}\right )}{\left (\frac {\left (-a \,b^{2}\right )^{\frac {1}{3}}}{2 b}+\frac {i \sqrt {3}\, \left (-a \,b^{2}\right )^{\frac {1}{3}}}{2 b}\right ) \left (\frac {3 \left (-a \,b^{2}\right )^{\frac {1}{3}}}{2 b}-\frac {i \sqrt {3}\, \left (-a \,b^{2}\right )^{\frac {1}{3}}}{2 b}\right )}}\right ) \sqrt {x \left (b \,x^{3}+a \right )}}{55 a \left (-\frac {3 \left (-a \,b^{2}\right )^{\frac {1}{3}}}{2 b}+\frac {i \sqrt {3}\, \left (-a \,b^{2}\right )^{\frac {1}{3}}}{2 b}\right ) \left (-a \,b^{2}\right )^{\frac {1}{3}} \sqrt {b x \left (x -\frac {\left (-a \,b^{2}\right )^{\frac {1}{3}}}{b}\right ) \left (x +\frac {\left (-a \,b^{2}\right )^{\frac {1}{3}}}{2 b}+\frac {i \sqrt {3}\, \left (-a \,b^{2}\right )^{\frac {1}{3}}}{2 b}\right ) \left (x +\frac {\left (-a \,b^{2}\right )^{\frac {1}{3}}}{2 b}-\frac {i \sqrt {3}\, \left (-a \,b^{2}\right )^{\frac {1}{3}}}{2 b}\right )}\, \sqrt {x}\, \sqrt {b \,x^{3}+a}}\) | \(745\) |
| elliptic | \(\frac {\sqrt {x \left (b \,x^{3}+a \right )}\, \left (-\frac {2 A \sqrt {b \,x^{4}+a x}}{11 x^{6}}-\frac {2 \left (3 A b +11 B a \right ) \sqrt {b \,x^{4}+a x}}{55 a \,x^{3}}+\frac {2 \left (B b -\frac {2 b \left (3 A b +11 B a \right )}{55 a}\right ) \left (\frac {\left (-a \,b^{2}\right )^{\frac {1}{3}}}{2 b}-\frac {i \sqrt {3}\, \left (-a \,b^{2}\right )^{\frac {1}{3}}}{2 b}\right ) \sqrt {\frac {\left (-\frac {3 \left (-a \,b^{2}\right )^{\frac {1}{3}}}{2 b}+\frac {i \sqrt {3}\, \left (-a \,b^{2}\right )^{\frac {1}{3}}}{2 b}\right ) x}{\left (-\frac {\left (-a \,b^{2}\right )^{\frac {1}{3}}}{2 b}+\frac {i \sqrt {3}\, \left (-a \,b^{2}\right )^{\frac {1}{3}}}{2 b}\right ) \left (x -\frac {\left (-a \,b^{2}\right )^{\frac {1}{3}}}{b}\right )}}\, {\left (x -\frac {\left (-a \,b^{2}\right )^{\frac {1}{3}}}{b}\right )}^{2} \sqrt {\frac {\left (-a \,b^{2}\right )^{\frac {1}{3}} \left (x +\frac {\left (-a \,b^{2}\right )^{\frac {1}{3}}}{2 b}+\frac {i \sqrt {3}\, \left (-a \,b^{2}\right )^{\frac {1}{3}}}{2 b}\right )}{b \left (-\frac {\left (-a \,b^{2}\right )^{\frac {1}{3}}}{2 b}-\frac {i \sqrt {3}\, \left (-a \,b^{2}\right )^{\frac {1}{3}}}{2 b}\right ) \left (x -\frac {\left (-a \,b^{2}\right )^{\frac {1}{3}}}{b}\right )}}\, \sqrt {\frac {\left (-a \,b^{2}\right )^{\frac {1}{3}} \left (x +\frac {\left (-a \,b^{2}\right )^{\frac {1}{3}}}{2 b}-\frac {i \sqrt {3}\, \left (-a \,b^{2}\right )^{\frac {1}{3}}}{2 b}\right )}{b \left (-\frac {\left (-a \,b^{2}\right )^{\frac {1}{3}}}{2 b}+\frac {i \sqrt {3}\, \left (-a \,b^{2}\right )^{\frac {1}{3}}}{2 b}\right ) \left (x -\frac {\left (-a \,b^{2}\right )^{\frac {1}{3}}}{b}\right )}}\, b \operatorname {EllipticF}\left (\sqrt {\frac {\left (-\frac {3 \left (-a \,b^{2}\right )^{\frac {1}{3}}}{2 b}+\frac {i \sqrt {3}\, \left (-a \,b^{2}\right )^{\frac {1}{3}}}{2 b}\right ) x}{\left (-\frac {\left (-a \,b^{2}\right )^{\frac {1}{3}}}{2 b}+\frac {i \sqrt {3}\, \left (-a \,b^{2}\right )^{\frac {1}{3}}}{2 b}\right ) \left (x -\frac {\left (-a \,b^{2}\right )^{\frac {1}{3}}}{b}\right )}}, \sqrt {\frac {\left (\frac {3 \left (-a \,b^{2}\right )^{\frac {1}{3}}}{2 b}+\frac {i \sqrt {3}\, \left (-a \,b^{2}\right )^{\frac {1}{3}}}{2 b}\right ) \left (\frac {\left (-a \,b^{2}\right )^{\frac {1}{3}}}{2 b}-\frac {i \sqrt {3}\, \left (-a \,b^{2}\right )^{\frac {1}{3}}}{2 b}\right )}{\left (\frac {\left (-a \,b^{2}\right )^{\frac {1}{3}}}{2 b}+\frac {i \sqrt {3}\, \left (-a \,b^{2}\right )^{\frac {1}{3}}}{2 b}\right ) \left (\frac {3 \left (-a \,b^{2}\right )^{\frac {1}{3}}}{2 b}-\frac {i \sqrt {3}\, \left (-a \,b^{2}\right )^{\frac {1}{3}}}{2 b}\right )}}\right )}{\left (-\frac {3 \left (-a \,b^{2}\right )^{\frac {1}{3}}}{2 b}+\frac {i \sqrt {3}\, \left (-a \,b^{2}\right )^{\frac {1}{3}}}{2 b}\right ) \left (-a \,b^{2}\right )^{\frac {1}{3}} \sqrt {b x \left (x -\frac {\left (-a \,b^{2}\right )^{\frac {1}{3}}}{b}\right ) \left (x +\frac {\left (-a \,b^{2}\right )^{\frac {1}{3}}}{2 b}+\frac {i \sqrt {3}\, \left (-a \,b^{2}\right )^{\frac {1}{3}}}{2 b}\right ) \left (x +\frac {\left (-a \,b^{2}\right )^{\frac {1}{3}}}{2 b}-\frac {i \sqrt {3}\, \left (-a \,b^{2}\right )^{\frac {1}{3}}}{2 b}\right )}}\right )}{\sqrt {x}\, \sqrt {b \,x^{3}+a}}\) | \(760\) |
| default | \(\text {Expression too large to display}\) | \(3690\) |
Input:
int((b*x^3+a)^(1/2)*(B*x^3+A)/x^(13/2),x,method=_RETURNVERBOSE)
Output:
-2/55*(b*x^3+a)^(1/2)*(3*A*b*x^3+11*B*a*x^3+5*A*a)/x^(11/2)/a-6/55*b^2*(2* A*b-11*B*a)/a*(1/2/b*(-a*b^2)^(1/3)-1/2*I*3^(1/2)/b*(-a*b^2)^(1/3))*((-3/2 /b*(-a*b^2)^(1/3)+1/2*I*3^(1/2)/b*(-a*b^2)^(1/3))*x/(-1/2/b*(-a*b^2)^(1/3) +1/2*I*3^(1/2)/b*(-a*b^2)^(1/3))/(x-1/b*(-a*b^2)^(1/3)))^(1/2)*(x-1/b*(-a* b^2)^(1/3))^2*(1/b*(-a*b^2)^(1/3)*(x+1/2/b*(-a*b^2)^(1/3)+1/2*I*3^(1/2)/b* (-a*b^2)^(1/3))/(-1/2/b*(-a*b^2)^(1/3)-1/2*I*3^(1/2)/b*(-a*b^2)^(1/3))/(x- 1/b*(-a*b^2)^(1/3)))^(1/2)*(1/b*(-a*b^2)^(1/3)*(x+1/2/b*(-a*b^2)^(1/3)-1/2 *I*3^(1/2)/b*(-a*b^2)^(1/3))/(-1/2/b*(-a*b^2)^(1/3)+1/2*I*3^(1/2)/b*(-a*b^ 2)^(1/3))/(x-1/b*(-a*b^2)^(1/3)))^(1/2)/(-3/2/b*(-a*b^2)^(1/3)+1/2*I*3^(1/ 2)/b*(-a*b^2)^(1/3))/(-a*b^2)^(1/3)/(b*x*(x-1/b*(-a*b^2)^(1/3))*(x+1/2/b*( -a*b^2)^(1/3)+1/2*I*3^(1/2)/b*(-a*b^2)^(1/3))*(x+1/2/b*(-a*b^2)^(1/3)-1/2* I*3^(1/2)/b*(-a*b^2)^(1/3)))^(1/2)*EllipticF(((-3/2/b*(-a*b^2)^(1/3)+1/2*I *3^(1/2)/b*(-a*b^2)^(1/3))*x/(-1/2/b*(-a*b^2)^(1/3)+1/2*I*3^(1/2)/b*(-a*b^ 2)^(1/3))/(x-1/b*(-a*b^2)^(1/3)))^(1/2),((3/2/b*(-a*b^2)^(1/3)+1/2*I*3^(1/ 2)/b*(-a*b^2)^(1/3))*(1/2/b*(-a*b^2)^(1/3)-1/2*I*3^(1/2)/b*(-a*b^2)^(1/3)) /(1/2/b*(-a*b^2)^(1/3)+1/2*I*3^(1/2)/b*(-a*b^2)^(1/3))/(3/2/b*(-a*b^2)^(1/ 3)-1/2*I*3^(1/2)/b*(-a*b^2)^(1/3)))^(1/2))*(x*(b*x^3+a))^(1/2)/x^(1/2)/(b* x^3+a)^(1/2)
Time = 0.15 (sec) , antiderivative size = 76, normalized size of antiderivative = 0.28 \[ \int \frac {\sqrt {a+b x^3} \left (A+B x^3\right )}{x^{13/2}} \, dx=-\frac {2 \, {\left (3 \, {\left (11 \, B a b - 2 \, A b^{2}\right )} \sqrt {a} x^{6} {\rm weierstrassPInverse}\left (0, -\frac {4 \, b}{a}, \frac {1}{x}\right ) + {\left ({\left (11 \, B a^{2} + 3 \, A a b\right )} x^{3} + 5 \, A a^{2}\right )} \sqrt {b x^{3} + a} \sqrt {x}\right )}}{55 \, a^{2} x^{6}} \] Input:
integrate((b*x^3+a)^(1/2)*(B*x^3+A)/x^(13/2),x, algorithm="fricas")
Output:
-2/55*(3*(11*B*a*b - 2*A*b^2)*sqrt(a)*x^6*weierstrassPInverse(0, -4*b/a, 1 /x) + ((11*B*a^2 + 3*A*a*b)*x^3 + 5*A*a^2)*sqrt(b*x^3 + a)*sqrt(x))/(a^2*x ^6)
Result contains complex when optimal does not.
Time = 73.18 (sec) , antiderivative size = 97, normalized size of antiderivative = 0.36 \[ \int \frac {\sqrt {a+b x^3} \left (A+B x^3\right )}{x^{13/2}} \, dx=\frac {A \sqrt {a} \Gamma \left (- \frac {11}{6}\right ) {{}_{2}F_{1}\left (\begin {matrix} - \frac {11}{6}, - \frac {1}{2} \\ - \frac {5}{6} \end {matrix}\middle | {\frac {b x^{3} e^{i \pi }}{a}} \right )}}{3 x^{\frac {11}{2}} \Gamma \left (- \frac {5}{6}\right )} + \frac {B \sqrt {a} \Gamma \left (- \frac {5}{6}\right ) {{}_{2}F_{1}\left (\begin {matrix} - \frac {5}{6}, - \frac {1}{2} \\ \frac {1}{6} \end {matrix}\middle | {\frac {b x^{3} e^{i \pi }}{a}} \right )}}{3 x^{\frac {5}{2}} \Gamma \left (\frac {1}{6}\right )} \] Input:
integrate((b*x**3+a)**(1/2)*(B*x**3+A)/x**(13/2),x)
Output:
A*sqrt(a)*gamma(-11/6)*hyper((-11/6, -1/2), (-5/6,), b*x**3*exp_polar(I*pi )/a)/(3*x**(11/2)*gamma(-5/6)) + B*sqrt(a)*gamma(-5/6)*hyper((-5/6, -1/2), (1/6,), b*x**3*exp_polar(I*pi)/a)/(3*x**(5/2)*gamma(1/6))
\[ \int \frac {\sqrt {a+b x^3} \left (A+B x^3\right )}{x^{13/2}} \, dx=\int { \frac {{\left (B x^{3} + A\right )} \sqrt {b x^{3} + a}}{x^{\frac {13}{2}}} \,d x } \] Input:
integrate((b*x^3+a)^(1/2)*(B*x^3+A)/x^(13/2),x, algorithm="maxima")
Output:
integrate((B*x^3 + A)*sqrt(b*x^3 + a)/x^(13/2), x)
\[ \int \frac {\sqrt {a+b x^3} \left (A+B x^3\right )}{x^{13/2}} \, dx=\int { \frac {{\left (B x^{3} + A\right )} \sqrt {b x^{3} + a}}{x^{\frac {13}{2}}} \,d x } \] Input:
integrate((b*x^3+a)^(1/2)*(B*x^3+A)/x^(13/2),x, algorithm="giac")
Output:
integrate((B*x^3 + A)*sqrt(b*x^3 + a)/x^(13/2), x)
Timed out. \[ \int \frac {\sqrt {a+b x^3} \left (A+B x^3\right )}{x^{13/2}} \, dx=\int \frac {\left (B\,x^3+A\right )\,\sqrt {b\,x^3+a}}{x^{13/2}} \,d x \] Input:
int(((A + B*x^3)*(a + b*x^3)^(1/2))/x^(13/2),x)
Output:
int(((A + B*x^3)*(a + b*x^3)^(1/2))/x^(13/2), x)
\[ \int \frac {\sqrt {a+b x^3} \left (A+B x^3\right )}{x^{13/2}} \, dx=\frac {2 \sqrt {b \,x^{3}+a}\, a -16 \sqrt {b \,x^{3}+a}\, b \,x^{3}+27 \sqrt {x}\, \left (\int \frac {\sqrt {x}\, \sqrt {b \,x^{3}+a}}{b \,x^{10}+a \,x^{7}}d x \right ) a^{2} x^{5}}{16 \sqrt {x}\, x^{5}} \] Input:
int((b*x^3+a)^(1/2)*(B*x^3+A)/x^(13/2),x)
Output:
(2*sqrt(a + b*x**3)*a - 16*sqrt(a + b*x**3)*b*x**3 + 27*sqrt(x)*int((sqrt( x)*sqrt(a + b*x**3))/(a*x**7 + b*x**10),x)*a**2*x**5)/(16*sqrt(x)*x**5)